The way to prevent this from happening is to get rid of the rose of Sharon. Or replace with one of the sterile forms.

A broadleaved weed killer should take care of them but it could take more than one application.

I agree. And as soon as you 'see' new ones, cut them off. Mine do not seem to regrow after I pull them out. Because they sometimes grow in unusual places, I suspect the birds are also helping propagate them. They eat the seeds and deposit them elsewhere for example. At first, I did not have any ROSs and then a few sprouted. The neighbors do not have any ROSs so birds may have delivered them. I finally added some ROSs that were sterile and those performed as expected.